The invention concerns an analog electronic circuit and calculation and demodulation method that solve a problem which appears in receivers of digital communications and elsewhere, especially in connection with analog decoders.
For example, consider a communication system that uses PAM (pulse amplitude modulation) signaling. Assuming AWGN (additive white Gaussian noise), the discrete-time channel between the modulator input in the transmitter and the matched-filter output in the receiver may be modeled asYk=h·Xk+Zk,  (1)where Xk is the (real-valued) transmitted symbol at time k, where h is a scale factor (attenuation or gain), and where Yk is the (real-valued) received symbol at time k. The transmitted symbol Xk is selected from a finite set S={s0, . . . ,sM−1} (a set of M real numbers). The noise process Z1,Z2, . . . is a sequence of independent zero-mean Gaussian random variables with variance σ2 which is independent of the input process X1,X2, . . . . To simplify the notation, we will drop the time index k and write (1) asY=h·X+Z.  (2)
For any choice of X=si, the conditional probability density of Y isfY|X(y|si)=(2πσ2)−1/2 exp(−(y−hsi)2/(2σ2))  (3)
In many types of receivers, it is necessary to compute the M numbers (likelihoods) fY|X(y|s0), . . . , fY|X(y|sM−1) for the (fixed) received symbol y, up to some common scale factor. In other words, we need to evaluate (3) for i=0, . . . ,M−1 (up to a scale factor). In particular, these M numbers (or their logarithms) are needed as input for decoders of turbo codes and similar codes.
For any fixed y, we can write (3) asfY|X(y|si)∝exp(−(y−hsi)2/(2σ2))  (4)∝exp((2yhsi−(hsi)2)/(2σ2)),  (5)where “∝” denotes equality up to a scale factor that does not depend on i (but may depend on y).
Now assume that the receiver contains an analog decoder for some error correcting code, e.g., as proposed in [USPatA], [LLHT:ppdavlsi2001], [Lus:davlsiid2000]. A main attraction of such decoders is that, in principle, no A/D (analog-to-digital) converter is needed. This requires, of course, that (3) (or something equivalent) is evaluated in the analog domain. For the special case M=2 (binary antipodal signaling), a simple circuit for this computation is shown in FIG. 1. The output of this circuit are two currents I0 and I1 with(I0, I1)∝(fY|X(y|+1), fY|X(y|−1)).
This circuit was presented in the talk [Lo:dead1999c], the slides of which have long been available on the web.
For M>2, however, the computation of (3) has been an open problem. An approximate solution for 16-QAM (quadrature amplitude modulation) was recently proposed in [SLLAJ:ad2004].